Learn Mathematics

Limit Laws

We will now look at some limit laws that will be important to apply in evaluating more difficult limits:

Law 1 (Direct Substitution Law and Limit Constant Law): If $f$ is a function defined at $x = a$, then the limit of $f$ equals the function $f$ evaluated at $a$, that is, $\lim_{x \to a} f(x) = f(a)$. Furthermore, if $f$ is a constant function $k$, then the limit of $f$ is equal to $k$, that is, $\lim_{x \to a} f(x) = f(k) = k$.

Law 2 (Limit Sum Law): If $f$ and $g$ are functions, then the limit of the sum $f + g$ is equal to the limit of $f$ summed with the limit of $g$, that is, $\lim_{x \to a}[ f(x) + g(x) ] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$.

Law 3 (Limit Difference Law): If $f$ and $g$ are functions, then the limit of the difference $f - g$ is equal to the limit of $f$ differenced with the limit of $g$, that is, $\lim_{x \to a}[ f(x) - g(x) ] = \lim_{x \to a} f(x) - \lim_{x \to a} g(x)$.

Law 4 (Limit Constant Multiple Law): If $f$ is a function and $k$ is some constant, then the limit of $k$-times $f$ is equal to $k$-times the limit of $f$, that is, $\lim_{x \to a} kf(x) = k \lim_{x \to a} f(x)$.

Law 5 (Limit Product Law): If $f$ and $g$ are functions, then the limit of the product $f \cdot g$ is equal to the limit of $f$ multiplied by the limit of $g$, that is $\lim_{x \to a} [f(x)g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$.

Law 6 (Limit Quotient Law): If $f$ and $g$ are functions, then the limit of the quotient $\frac{f}{g}$ is equal to the limit of $f$ divided by the limit of $g$, that is $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$ provided that $\lim_{x \to a} g(x) \neq 0$.

Law 7 (Limit Power Law): If $f$ is a function and $n \in \mathbb{Z}^+$, then the limit of $f^n$ is equal to the limit of $f$ all raised to the $n$th power, that is $\lim_{x \to a} [f(x)]^n = \left (\lim_{x \to a} f(x) \right )^n$.